Let x and y be non-zero, rational numbers and let z be an irrational number. In the first column of the table below is an expression. In the second column, determine whether the value of each expression is rational or irrational.
Expression Value of Expression
x + y
y + z
xy
xz
Expression Value of Expression
x + y Rational
y + z Irrational
xy Rational
xz Irrational
To determine whether the value of each expression is rational or irrational, we need to understand the properties of rational and irrational numbers.
Rational numbers are those that can be expressed as a ratio of two integers, while irrational numbers cannot be expressed as a ratio and have an infinite non-repeating decimal representation.
Now let's analyze each expression:
1. x + y:
Since x and y are given to be non-zero rational numbers, their sum (x + y) will also be a rational number. Therefore, the value of this expression is rational.
2. y + z:
Here, y is a non-zero rational number, and z is an irrational number. The addition of a rational and an irrational number will always result in an irrational number. Therefore, the value of this expression is irrational.
3. xy:
As both x and y are non-zero rational numbers, their product (xy) will also be a rational number. Therefore, the value of this expression is rational.
4. xz:
Since x is a non-zero rational number and z is an irrational number, the product of a rational and an irrational number will always be an irrational number. Therefore, the value of this expression is irrational.
To summarize:
Expression | Value of Expression
--------------------------------------
x + y | Rational
y + z | Irrational
xy | Rational
xz | Irrational
To determine whether the value of each expression is rational or irrational, we need to understand the properties of rational and irrational numbers.
Rational numbers can be expressed as fractions, where the numerator and denominator are both integers, and the denominator is not zero. For example, 3/4, -2/5, and 7 are all rational numbers.
Irrational numbers, on the other hand, cannot be expressed as a simple fraction. Examples of irrational numbers include √2, π, and e.
Now, let's analyze each expression:
1. x + y:
Since x and y are given to be non-zero rational numbers, their sum will also be rational. The sum of any two rational numbers is always rational.
2. y + z:
Here, y is a non-zero rational number and z is an irrational number. When we add a rational number and an irrational number, the result will always be irrational. Therefore, the value of this expression is irrational.
3. xy:
When we multiply two rational numbers, the result is again a rational number. Therefore, the value of this expression is rational.
4. xz:
In this expression, we are multiplying a rational number (x) and an irrational number (z). Multiplying a rational number with an irrational number results in an irrational number. Therefore, the value of this expression is irrational.
To summarize:
Expression | Value of Expression
----------------------------------------
x + y | Rational
y + z | Irrational
xy | Rational
xz | Irrational